First, is it even possible to state a well known problem and know what axioms are allowed for it? I'd guess that one must look at the branches of math involved, but I'm no mathematician.
Therefore, I would like to ask the people working on this problem if this there is a fixed list of allowed axioms for this problem.
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Naively, the answer is "any good ol' axiomatic system which is not obviously inconsistent and can provide a sound basis for arithmetic".
So you can work with Peano Arithmetic, or the second-order Peano axioms, or one of many set theories, or just $\sf ZFC$, or augment $\sf ZFC$ by any large cardinal axiom, or by axioms like $V=L$, and much more.
The thing here, is that almost none of this matters. The Riemann Hypothesis is equivalent to a $\Sigma_1$ sentence in the language of arithmetic, let's call it $\varphi$ for now. This is an almost irrelevant fact, but it has an interesting consequence: it is enough to prove that $\varphi$ holds for the natural numbers in order to prove the hypothesis holds.
And it almost doesn't matter what is the axiomatic system you're using, since most of the basic axiomatic systems that allow development of arithmetic will agree on whether or not a $\Sigma_1$ sentence is true in $\Bbb N$.
(There is a caveat here, that these systems need to agree on what is $\Bbb N$ to begin with, but let's ignore this for now, because we can, and even a proof under this assumption would be extraordinary and probably accepted as a proof by the bulk of mathematicians.)
Most of everyday mathematics works with a set of axioms known as ZFC. Some mathematicians study possible (small) changes to that axiom system, with various interesting (to them) implications. But most of the alternatives proposed do not affect work on most open problems.
Here's an analogy. There are (probably) engineers who spend time thinking about ways to formulate concrete to make it a little cheaper, or a little stronger, or a little longer lasting. But what they do does not affect the design of skyscrapers, which will be stable with foundations built with the current industry standard concrete, and would be with any of the known acceptable variants. So architects don't really have to pay attention to the concrete.
Rarely. from time to time, there's a building design challenge where the construction does depend on those variants in the foundation.
I think most mathematicians think that the Riemann hypothesis is analogous to designing a really tall skyscraper. New construction techniques will be required, but new foundations won't.